Since it's generally okay to soak (many) pen parts in a mild ammonia solution in order to clean out residual ink, is there any reason one shouldn't simply fill the pen with this same mild ammonia solution and let it sit for a couple of days, then flush as normal? It seems like it would be an easy way to reach most of the places ink would creep into without needing to disassemble the pen...

I honestly don't know. I dont risk putting barrels in the ammonia/water mixture in case it deforms the pattern. In theory putting it in the pen would allow one to clean their pens without worry of damage. However I might still dissasemble the pen to help it dry.